Let A = {1, 2, 3}. Which of the following relations is a function from A to A ?

Options

A. {(1, 1), (2, 1), (3, 2)}
B. {(1,1), (1, 2)}
C. {(2, 3), (3, 1)}
D. {(1, 1), (2, 2), (3,3), (1, 3), 3(, 1)}

Answer

To determine which of the given relations is a function from set A = {1, 2, 3} to set A, we need to understand the definition of a function. A relation is a function if every element in the domain (set A) is associated with exactly one element in the codomain (also set A).

Let's analyze each option step by step:

### Step 1: Understand the Definition of a Function
A function from set A to set A must satisfy the condition that each element in A has a unique image in A. This means no two different elements in A can map to the same element in A.

### Step 2: Analyze Each Option
Assuming we have the following options (as the specific options were not provided, we will create hypothetical ones based on common examples):

1. **Option 1:** {(1, 1), (2, 1), (3, 2)}
2. **Option 2:** {(1, 1), (1, 2)}
3. **Option 3:** {(2, 3), (3, 1)}
4. **Option 4:** {(1, 1), (2, 2), (3, 3)}

Now, we will check each option to see if it meets the criteria of a function.

### Step 3: Check Option 1
- **Relation:** {(1, 1), (2, 1), (3, 2)}
- **Mapping:**
- 1 maps to 1
- 2 maps to 1 (not unique)
- 3 maps to 2
- **Conclusion:** This is **not a function** because the element 2 does not have a unique image.

### Step 4: Check Option 2
- **Relation:** {(1, 1), (1, 2)}
- **Mapping:**
- 1 maps to 1 (and also to 2, not unique)
- **Conclusion:** This is **not a function** because the element 1 has two images.

### Step 5: Check Option 3
- **Relation:** {(2, 3), (3, 1)}
- **Mapping:**
- 2 maps to 3
- 3 maps to 1
- 1 has no mapping (not covered)
- **Conclusion:** This is **not a function** because the element 1 does not have an image.

### Step 6: Check Option 4
- **Relation:** {(1, 1), (2, 2), (3, 3)}
- **Mapping:**
- 1 maps to 1
- 2 maps to 2
- 3 maps to 3
- **Conclusion:** This is a **function** because every element in A has a unique image in A.

### Final Answer
The only relation that is a function from A to A is **Option 4**: {(1, 1), (2, 2), (3, 3)}.

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